Abstract Algebra Theory and Applications - Computer Science ...

14.3 RING HOMOMORPHISMS AND IDEALS 241ar ∈ 〈a〉 by an arbitrary element s ∈ R, we have s(ar) = a(sr). Therefore,〈a〉 satisfies the definition of an ideal.If R is a commutative ring with identity, then an ideal of the form 〈a〉 ={ar : r ∈ R} is called a principal ideal.Theorem 14.7 Every ideal in the ring of integers Z is a principal ideal.Proof. The zero ideal {0} is a principal ideal since 〈0〉 = {0}. If I is anynonzero ideal in Z, then I must contain some positive integer m. Thereexists at least one such positive integer n in I by the Principle of Well-Ordering. Now let a be any element in I. Using the division algorithm, weknow that there exist integers q and r such thata = nq + rwhere 0 ≤ r < n. This equation tells us that r = a − nq ∈ I, but r must be0 since n is the least positive element in I. Therefore, a = nq and I = 〈n〉.□Example 17. The set nZ is ideal in the ring of integers. If na is in nZ andb is in Z, then nab is in nZ as required. In fact, by Theorem 14.7, these arethe only ideals of Z.Proposition 14.8 The kernel of any ring homomorphism φ : R → S is anideal in R.Proof. We know from group theory that ker φ is an additive subgroup ofR. Suppose that r ∈ R and a ∈ ker φ. Then we must show that ar and raare in ker φ. However,φ(ar) = φ(a)φ(r) = 0φ(r) = 0andφ(ra) = φ(r)φ(a) = φ(r)0 = 0.□Remark. In our definition of an ideal we have required that rI ⊂ I andIr ⊂ I for all r ∈ R. Such ideals are sometimes referred to as two-sidedideals. We can also consider one-sided ideals; that is, we may requireonly that either rI ⊂ I or Ir ⊂ I for r ∈ R hold but not both. Suchideals are called left ideals and right ideals, respectively. Of course,in a commutative ring any ideal must be two-sided. In this text we willconcentrate on two-sided ideals.